Carnegie Mellon Electricity Industry Center Working Paper CEIC-07 www.cmu.edu/electricity 1
A Centrality Measure for Electrical Networks
Paul Hines and Seth Blumsack
types of failures. Many classifications of network structures Abstract—We derive a measure of “electrical centrality” for have been studied in the field of complex systems, statistical AC power networks, which describes the structure of the mechanics, and social networking [5,6], as shown in Figure 2, network as a function of its electrical topology rather than its but the two most fruitful and relevant have been the random physical topology. We compare our centrality measure to network model of Erdös and Renyi [7] and the “small world” conventional measures of network structure using the IEEE 300- bus network. We find that when measured electrically, power model inspired by the analyses in [8] and [9]. In the random networks appear to have a scale-free network structure. Thus, network model, nodes and edges are connected randomly. The unlike previous studies of the structure of power grids, we find small-world network is defined largely by relatively short that power networks have a number of highly-connected “hub” average path lengths between node pairs, even for very large buses. This result, and the structure of power networks in networks. One particularly important class of small-world general, is likely to have important implications for the reliability networks is the so-called “scale-free” network [10, 11], which and security of power networks. is characterized by a more heterogeneous connectivity. In a Index Terms—Scale-Free Networks, Connectivity, Cascading scale-free network, most nodes are connected to only a few Failures, Network Structure others, but a few nodes (known as hubs) are highly connected to the rest of the network. The signature property of a scale- I. INTRODUCTION free network is a power-law distribution (a very fat-tailed N bulk electric power networks, wide-scale cascading distribution) of node connectivity or degree. Ifailures happen more often than would be expected if failures were random and independent, and their sizes followed a normal distribution [1]. Despite reliability standards and significant technological advances, the frequency of large blackouts in the United States has not decreased since the creation of NERC in 1965 [2]. The US blackout of August 2003, along with the Italian blackout of September 2003, convinced even skeptics that fundamental weaknesses exist in transmission infrastructures. Many have sought to explain this weakness as a function of a shift in electricity industry structure from regulated, vertically integrated utilities with moderate interregional trade, to a diverse set of market participants using the transmission infrastructure to facilitate long distance energy transactions [3,4]. While the merits and shortcomings of electricity market restructuring are a contentious subject, at the very least it is fair to say that the number and size of blackouts on the North American electric grid has not decreased in recent years, as Figure 1: The number of large blackouts according to [NERC DAWG] shown in Figure 1. between 1984 and 2006. The bars show the data after adjusting for demand growth (using year 2000 as the base year). The line shows the number of Recent advances in network and graph theory have drawn blackouts >400 MW before this adjustment. Clearly, the frequency of large links between the topological structure of networks cascading failures in the North American electric power system has not (particularly networks consisting of social ties and decreased during the 23 year period shown here. Assuming that large failures infrastructures) and the vulnerability of networks to certain were not more common before this period, the overall frequency of large blackouts has not decreased since the formation of NERC in 1965.
This work was supported in part by the Alfred P. Sloan Foundation and the Electrical Power Research Institute through grants to the Carnegie Mellon The structural differences between the random and scale- Electricity Industry Center and in part by the US Dept. of Energy through free network models have important implications for network grants to the National Energy Technology Laboratory. vulnerability. In particular, random networks tend to be fairly Paul Hines is with the School of Engineering, University of Vermont, Burlington, VT 05405 (e-mail: [email protected]). robust to targeted attacks but relatively vulnerable to a series Seth Blumsack is with the Dept. of Energy and Mineral Engineering, of random failures or attacks. Scale-free networks, on the Pennsylvania State University, University Park, PA 16902 (e-mail: other hand, are highly vulnerable to targeted attacks or failures [email protected]). at one of the hubs, but are more robust to failures at randomly Carnegie Mellon Electricity Industry Center Working Paper CEIC-07 www.cmu.edu/electricity 2 chosen nodes [34]. Similarly, a network’s structure has Section VI offers some conclusions and extensions. important implications for strategies aimed at protecting the network against cascading failures or more deliberate attacks.2 II. NETWORK CONNECTIVITY METRICS AND APPLICATIONS TO The world wide web is a good example of a scale-free POWER NETWORKS network; the network as a whole can be made more robust by Network connectivity is often measured using the degree increasing the reliability of a fairly small set of critical hubs of nodes in the network – the number of edges (and thereby (such as google.com or yahoo.com). other nodes) connected to a given node. The degree, k, distribution varies substantially from one network structure to another. In a regular lattice structure (such as the nearest neighbor and second neighbor graphs shown in Figure 2) k is constant for all nodes. Random networks have an exponential degree distribution, as do most small-world networks. When a small-world network shows a power-law degree distribution, rather than an exponential, it is known as a scale-free network [11]. If P(k) is the probability that a randomly chosen node has degree k, the degree distribution of a scale free network follows Pk()~ k−γ . In most real networks, γ falls in the range of 2 to 3. Many seemingly different types of networks have been found to possess a scale-free structure, including the world wide web, airline networks, and protein interactions in yeast [5]. Power networks represent a natural test case for complex Figure 2: Small examples of the network structures used in this paper. systems and network theory. However, the lack of widely The principal shortcoming of existing attempts to measure available data on the actual topology of the power grid, the network structure of electrical grids lies in the failure to particularly in the United States, has dramatically limited the explicitly incorporate the physical laws governing the flow of number of analyses. The few existing studies generally find electricity within these networks. While the topological (node- that power networks exhibit some kind of nonrandom edge) structure of an electrical power network may suggest structure, but there is disagreement over the actual structure of one set of behaviors and vulnerabilities, the electrical structure the network. In particular the existing studies disagree with of the network may suggest something altogether different. respect to whether the tail of the degree distribution follows a The flow of power through the network is governed by power law or exponential distribution. Portions of the North Kirchoff’s Laws and not simply by topology. To address this American grid have been discussed in [5, 6, 16, 18 – 20]. The shortcoming we introduce an electrical connectivity metric North American grid as a whole is discussed in [18], in which which incorporates the information contained in the system the authors report that the degree distribution of the power impedance matrix. This metric is referred to here as “electrical grid has an exponential or “single-scale” [11] form with a fast- centrality.” Electrical centrality accommodates Kirchoff’s decaying tail, although the distribution of the number of Laws and is more accurately captures the properties of node transmission lines passing through a given node (the centrality, relative to metrics based on node-edge connectivity. “betweenness” of a given node) does follow a power law. The This paper is organized in five sections. Section I is this authors of [16] find the same single-scale structure in the introduction, and section II reviews the existing literature degree distribution for a portion of the electric network in related to the subject matter. Section III reviews the classical California. The same result appears in [5] and [6] for the network connectivity metrics, which provide information only power grid of the Western United States. However, [10, 17] on the topological structure of the network in question, and estimate a power-law relationship in the degree distribution discusses some existing applications of these metrics to power for both the Western [10, 17] and Eastern [17] portions of the networks. Section IV introduces the electrical centrality U.S. power grid. metric. In Section V we use the IEEE 300-bus network to Structural properties of the Italian, French, and Spanish illustrate the differences between our electrical connectivity power networks can be found in [19] and [20]. These metric and the classical topological connectivity metrics. European power networks are generally found to have the same single-scale topological structure as the North American 2 It is important to realize that, particularly in complex systems such as grid, with an exponential tail in the degree distribution. The power networks, building more is not always better. The North American distribution of nodal demands (or loads), however, follows a blackout of August 2003 was followed by a number of calls for massive investments in transmission infrastructure; [12], for example, suggested a power law: most buses in the grid serve lower-than average program of investment and grid modernization amounting to $100 billion. The demand, but a small number of buses serve a relatively large analysis of [13] and [14] suggests that simply building more transmission lines amount of demand. is unlikely to meet reliability goals, and [15] suggests that more intelligent control strategies can improve reliability at lower costs. While some of the topological studies of power networks Carnegie Mellon Electricity Industry Center Working Paper CEIC-07 www.cmu.edu/electricity 3 simply sought to tease out information about the structure of 2. Power networks are best described by undirected graphs. the network (as in [5]), some papers focus on the relationship The model in [17] argues that the power network can be between network structure and system reliability. For the thought of as a directed graph associated with a particular North American power grid, [16] introduces the concept of state of the network, and small perturbations in the system “connectivity loss,” which measures the change in the ability are unlikely to change the directionality. That is, every of the network to deliver power following the disconnection state of the network can be mapped into a set of of a substation from the network.3 They find that the directional power flows. This does not hold true for many cases in a power network. Due to the highly non-linear connectivity loss is significantly larger when nodes 4 representing hubs are disconnected from the network, relative and generally non-convex nature of the equations that govern power flow in AC electrical systems, it is to the removal of random nodes. This suggests that the power possible, and in some cases quite likely, that small grid has some kind of scale-free structure, even if it does not changes in the state of the network can reverse the appear directly in the degree distribution. A similar attack direction of power flow along a particular path. vulnerability is shown in [18] for the Nordic power network 3. Because flows are governed by Kirchoff’s laws, to the and that of the Western U.S. In [17] the authors construct a extent that power networks are characterized by a hub probabilistic model of cascading failures as a function of the structure, there is no a priori way to tell whether the hubs exponent γ in the power law distribution and the number of represent large inexpensive generators, load pockets, nodes with degree one. They find that the loss of load substation tie-points, or some other physical structure probability (LOLP) predicted by the model for the Eastern and (this assumption is made in [16]).5 That is, a simple Western Interconnects is similar to the LOLP from actual model of preferential attachment, as in [10], is unlikely to reliability studies performed by the Bonneville Power be a good evolutionary model explaining the emergent Administration. structure of power networks.
III. AN ALTERNATIVE CONNECTIVITY METRIC The existing literature on the structure and vulnerability of electrical power networks largely takes a topological approach to measure distance and connectivity. That is, the power grid is described as a simple graph of nodes connected by edges. Vulnerability analyses focus on the physical islanding of portions of the grid through attacks or failures at multiple nodes. The topological approach paints a somewhat confusing picture of power networks. Referencing the IEEE 300-bus network shown in Figure 3, it is apparent that while there may be a few nodes in the network that fall outside of an exponential probability distribution (and thus some evidence of a power-law tail in the degree distribution), the power network lacks the strong hub structure that is apparent in scale-free networks such as the world wide web and some airline networks. We argue that a fundamental weakness of the existing Figure 3: Topology of the IEEE 300-bus test system. structural studies of power grids is that they focus on topological connectivity, while ignoring the electrical Of these, the most basic structural modeling issue is somehow connectivity. In particular, a reasonably complete incorporating Kirchoff’s Laws into the network structure characterization of the network structure of the power grid metrics. Power grids tend to have highly meshed structures, so would need to incorporate the following properties of the mere topological proximity between two nodes will, in electrical systems: many circumstances, have less of an impact on the 1. Flow in a power network is governed by Kirchoff’s Laws performance of the network than electrical proximity, which and not by decisions made by individual actors at may or may not correspond to topological distance. The individual nodes. Power injections propagate through the weighted degree distributions discussed in [22] represent a network following a path of least resistance, apportioned step towards resolving this measurement issue, but in the by the relative complex impedance of each equivalent analysis of [22] the weights are based on capacity constraints, path. not any kind of path admittance or law of motion.
3 The data set used in [16] cannot identify with certainty which nodes 4 represent generator substations, and which nodes represent connection points [21] offers evidence that convexity of the set of feasible solutions to the to the distribution network or tie-points within the high-voltage transmission AC power flow problem is rarely, if ever, achieved. 5 network. The authors make some assumptions regarding which nodes Some information may be contained in the network admittance matrix, as represent each type of substation. discussed below. For example, transformers tend to have much larger admittances than other pieces of network equipment. Carnegie Mellon Electricity Industry Center Working Paper CEIC-07 www.cmu.edu/electricity 4
The degree distribution is but one centrality measure use the information contained in the network impedance ranking the importance of given nodes in the network. matrix to calculate equivalent electrical distances between Another metric, more popular in the analysis of social pairs of nodes, and examine the properties of the graph based networks than physical or technological networks, is on this distance metric compared to the usual topological betweenness [23, 24], defined in [25] as the fraction of distance metrics. Second, we construct electrical networks to shortest paths between pairs of nodes that happen to pass conform to the properties of three common network models, through a given node. In the most basic sense, betweenness in and compare the electrical properties of these networks with and of itself suffers from the same modeling issue as the those of an actual network of a similar size. degree distribution – it considers the topological structure of the network, but not the pattern of network flow propagation. This shortcoming has been recognized in [25] through the use of an electrical circuit as an example of how the shortest path (which would include nodes with a high betweenness) may not carry the largest amount of information (or current, in this particular case). The circuit example is then generalized to discuss a broader class of betweenness measures based on random walks through networks. Social network analysts have introduced a measure known as information centrality to describe how information is passed through a network of associates [26, 27]. Operationally, this is similar to gossip coming through the grapevine. The information centrality associated with a given node (or actor or agent, in the specific case of social networks), describes how much of the network flow travels Figure 4: A node-branch representation of the IEEE 300-bus test system. along each path beginning or ending with that node. Expanding this metric to physical networks, information centrality is (at least conceptually) very closely related to the power transfer distribution factor (PTDF) matrix [28], in which the i,jth element indicates the change in power flow on the jth transmission line from a marginal change in net power injection at the ith bus. Flow-based betweenness metrics were originally considered in [29], with the circuit analysis of [25] an interesting variation. The authors of [30] demonstrate that if information propagates through the network following Kirchoff’s Laws, and if the network can be described as a single-commodity network (that is, there are single points of injection and withdrawal), then the resulting betweenness metric based on electrical flow is identical to the information centrality.6 Figure 5: The probability mass function (histogram) for node degree in the IEEE 300 bus network. IV. ARE POWER NETWORKS SCALE-FREE? That betweenness is equivalent to information centrality in certain circumstances is an interesting result. However, real power networks have multiple generators and loads, and most likely experience significant loop flows. Thus, the use of standard network metrics to compare power systems to standard network models is likely to be misleading. We examine the differences between power networks and other generic networks models using two novel methods. First, we
6 This would seem to confirm that the information centrality is equivalent to the power transfer distribution matrix for the special case of a power network with one (net) generator and one (net) load (a single-commodity network). As shown in [31], it is possible to create an equivalent single- commodity power network out of a multi-commodity network only if there are Figure 6: Based on the Watts-Strogatz clustering coefficient, the IEEE 300- no loop flows. Thus, this special case is probably not all that interesting. bus test system shares the structural complexity of much larger electric power networks. Carnegie Mellon Electricity Industry Center Working Paper CEIC-07 www.cmu.edu/electricity 5
⎧ GjBklkl+≠ kl The test case for this work is the IEEE 300-bus test bus ⎪ Y = (1) kl ⎨− network, which has 300 buses (nodes) and 411 branches ∑()GjBklkl+= kl (transmission lines or transformers). Figure 4 shows the ⎩⎪ kl≠ topology of this network, while Figure 5 shows its degree bus distribution. The 300-bus network is much smaller than most The definition of the Y matrix used here captures both the real power grids, which may have tens, if not hundreds of real and reactive (imaginary) portions of the line admittances. bus bus thousands of buses and transmission lines. However, the 300- The Y matrix tends to be sparse, since Ykl = 0 for pairs of bus system is sufficiently large so as to share several structural nodes k and l that do not share a direct physical connection. In properties with actual power grids. An example is shown in our analysis we actually use the inverse of the Ybus matrix, Figure 6, which shows the Watts-Strogatz [9] clustering which is traditionally denoted the Zbus (or impedance) matrix. coefficient for standard IEEE test systems of various sizes, as Inverting the sparse Ybus matrix for a fully connected electrical well as the clustering coefficient for the power grid of the power network, yields a non-sparse (dense) matrix, denoted bus Western U.S., as reported in [9, 15]. While the 300-bus Z . The equivalent electrical distance between nodes k and l system is smaller than actual power networks, it is sufficiently is thus given by the magnitude of the relevant entry of the Zbus interesting for our analysis. bus matrix. Smaller ||Zkl correspond to shorter electrical A. Structural Properties of the 300-Bus Network distances (and a larger propensity for power to flow between The topological representation of the IEEE 300-bus these nodes, subject to capacity constraints along any of the network in Figures 4 and 5 can be visually compared with topological paths). similarly-sized small-world and scale-free networks, as shown In Figure 8 we redraw the 300-bus network to show the in Figure 7. The representation of the IEEE 300-bus network electrical connections rather than the topological connections. There are [(3002)/2 – 300] = 44,700 distinct node-to-node shown in Figure 4 does not immediately reveal any highly- bus connected hubs that would suggest a scale-free network connections in the Z matrix for the 300-bus network; the structure. Similarly, the degree distribution of the 300-bus links shown in Figure 8 represent the 411 node pairs that have the largest electrical connections. The threshold value for the network (Figure 5) does not fit well with a power law statistic. bus Further, the geodesic paths between random node pairs are entries of the Z matrix is 0.00255, as shown in Figure 9. longer than would be suggested by a Watts-Strogatz small- Thus, Figure 8 displays a graph approximately the same size world structure, where the average geodesic path length is less as the original network topology shown in Figure 3 (300 than six, even for very large networks. The topological nodes and 411 edges). The atomistic nodes show buses in the structure of the 300-bus network would lead us to agree with network that have small electrical connections to the network the analysis in [5, 16, 18] that power grids are not as a whole. characterized by a scale-free structure.
Figure 7: Topological representation of the IEEE 300-bus network (bottom center) as compared with a similarly-sized random network (upper right) and scale-free network (upper left).
However, as noted above and in [25, 30], flow in electrical networks is governed by Kirchoff’s Laws, which result in unique patterns of interaction between nodes in a network. The topological structure of the power network may therefore not say very much regarding the behavior of the network, and particularly its vulnerabilities. Kirchoff’s Laws are captured in the system bus-bus admittance matrix, defined by: Figure 8: The IEEE 300-bus network has been redrawn here to highlight the structure of the electrical connections represented in the Zbus matrix. To make this representation size-compatible with the topological representation in Figure 3, only the 411 strongest electrical connections (out of 44,700 total electrical connections) are shown. Carnegie Mellon Electricity Industry Center Working Paper CEIC-07 www.cmu.edu/electricity 6
bus The two representations of the 300-bus network suggest Based on the electrical information contained in the Z very different structures. From a topological perspective, the matrix, we re-interpret the degree distribution for the 300-bus 300-bus power network looks neither like a random network network to show the electrical betweenness of each bus in the nor a scale-free network. But from an electrical perspective system. The electrical degree distribution is shown in Figure 11. (which captures the behavior of the network, not simply the physical structure), the 300-bus network looks to have a distinct group of nodes that are “electrical hubs” – that is, buses that are have a high electrical connectivity to the rest of the network. Power flowing through the network is, due to Kirchoff’s Laws, much more likely to pass through these nodes than other nodes. In the language of social networks, these would correspond to nodes with a high betweenness or information centrality. Figure 10 redraws the topological representation of the 300-bus network, but with the node sizes adjusted according to the electrical centrality (nodes with a higher electrical centrality are shown larger in the figure).
Figure 11: The inverse cumulative probability density function for the electrical betweenness (electrical degree) of the nodes in the IEEE 300-bus network. B. Comparison of the IEEE 300 Bus Network with Other Network Structures. Topologically, the 300 bus power network does not have the properties of a scale-free network. Electrically, however, there appear to be a number of highly-connected nodes similar to what would be expected from a scale-free network. We now extend this analysis by creating simple electrical networks from the network topologies shown in figures 2, 4 and 7. Specifically, we construct simple network of resistors Figure 9: The inverse cumulative probability distribution function for all corresponding to the topologies of the nearest-neighbor, node-to-node electrical connections. The threshold at which only 411 stronger random, and scale-free networks in addition to the 300-bus bus (smaller ) connections exist is 0.00255, as indicated. ||Zlk power network. To build the resistor networks, at each node we place a 1 Ω resistor and represent each link by a 1 Ω resistor. For each network, we calculate a resistance matrix, which is essentially equivalent to the Ybus matrix for an equivalent power network. The resistance matrix R, defines the sensitivity relationship between voltages and currents, as shown in eq. (2).
ΔVR= ΔI , (2)
Given this relationship, we can calculate a sensitivity matrix S according to:
SRRkl= kl/ kk for all k and l. (3)
The elements of S, describe the extent to which a state change at location i will affect a similar change at location j. Figure 10: A representation of the IEEE 300-bus network, with the node-sizes Essentially this matrix tells the extent to which information adjusted to represent the relative magnitudes of the electrical betweenness (or whatever it is that is flowing through the system) will measure. The nodes in the center with high electrical connectivity (critical nodes in the network) can clearly be seen. propagate through each network, or conversely, the extent to Carnegie Mellon Electricity Industry Center Working Paper CEIC-07 www.cmu.edu/electricity 7 which information can be contained within a small area. In all C. Extensions: Implications for Vulnerability five networks, sensitivity decays approximately linearly with Network structure is closely related to network the distance between nodes i and j (see Figure 12), but vulnerability [16]. Thus, a deeper understanding of the because of the high connectivity of the hubs in the scale free structure of electric power grids can yield important insights networks, and the relatively short distances between nodes in for decreasing vulnerability and improving reliability. the random network, the number of nodes that would be Presently, the authors are looking at the relationship between affected by a given change is dramatically different for the the proposed electrical centrality metric and the vulnerability different network structures (see Figure 13). In other words, if of the system to failure and attack at particular locations. a scale-free or random network has properties similar to While the outcome of this research is pending, it is clear that a Kirchoff’s Laws, it will be very difficult to contain the complex-networks perspective can provide some insight into propagation of information to a small area. The power the vulnerability of electrical power networks. network has substantially more information propagation than that shown by the second neighbor lattice structure, but much V. DISCUSSION AND EXTENSIONS less than the random and scale free test cases. A clear link exists between the topological structure of many networks and types of failures to which they are vulnerable or robust. Scale-free networks are particularly vulnerable to failures at their highly connected hubs. This makes them particularly susceptible to deliberate attacks on these hubs, but less vulnerable to failures at random locations. A series of denial-of-service attacks targeting a major hub in the world wide web (such as Google or Yahoo) could cause ripple effects crippling much of the physical internet [35]. The winter of 2006/2007 proved particularly difficult for airlines and their passengers, as snow and ice storms hitting hub airports caused major delays in areas where the weather was perfect. This suggests that resources aimed at protecting scale- free networks should primarily be directed towards strengthening the hubs. The North American power blackout in August of 2003 has focused attention on network reliability and measures which could limit or prevent cascading failures. Many of the Figure 12: Average relative sensitivity Si,j plotted against the distance between i and j, for networks of resistors with different structures. For a given physical suggested measures, such as [12] involve large capital distance, the sensitivity is generally larger in the IEEE 300-bus network than expenditures on new infrastructure. However, the nature of in random or scale-free networks of approximately the same size. Information power networks is such that simply building lots of new thus propagates more slowly in the 300-bus network than it would if the network had a scale-free or random structure. transmission lines may not yield the desired performance improvements [13]. Further, it is not obvious that building more centrally-controlled infrastructure is necessarily a more cost-effective solution than implementing other novel control strategies [15]. Most analyses of the power grid have found a single-scale topological structure, which can arise if linking to high- connectivity nodes is sufficiently costly to impair preferential attachment in the growth of the network [11]. Due the cost of building new links (transmission lines) simply increasing the connectivity of the grid (as in [12]) is prohibitively expensive. Further, siting of new transmission lines is becoming increasingly difficult [32]; even with the possibility of federal intervention [33] it simply may not be feasible to expand the current network in a fashion which optimizes its resilience. Our analysis shows that, measured properly, power networks bear some resemblance to scale-free networks. The Figure 13: Distribution of the sensitivities for the networks of resistors. The hubs do not show up in a simple analysis of a power quantity of a network affected by some change or disturbance depends on (or network’s topology, but a more detailed look at the network’s varies with) the structure of the network. electrical structure reveals a network that shares many properties with other scale-free networks. The hubs in power networks themselves appear highly connected with other hubs, Carnegie Mellon Electricity Industry Center Working Paper CEIC-07 www.cmu.edu/electricity 8 indicating the presence of a highly vulnerable “core.” Further, [22] Barrat, A., M. Barthelemy, R. Pastor-Sartorras, and A. Vespignani, disturbances in one part of an electrical network affect local 2004. “The Architecture of Complex Weighted Networks,” Proceedings of the National Academy of Sciences 101:11, pp. 3747 – 3752. areas substantially more than they affect remote regions. The [23] Freeman, L. C., 1977. “A Set of Measures of Centrality Based Upon effects are spread less widely in the power network relative to Betweenness,” Sociometry 40, pp. 35 – 41. some other network structures. [24] Freeman, L. C., 1979. “Centrality in Social Networks: Conceptual Clarification,” Social Networks 1, pp. 215 – 239. [25] Newman, M. E. J., 2005. “A Measure of Betweenness Centrality Based VI. ACKNOWLEDGMENT on Random Walks,” Social Networks 27, pp. 39 – 54. [26] Stephenson, K. and M. Zelen, 1989. “Rethinking Centrality: Methods The authors would like to thank Sarosh Talukdar for helpful and Applications,” Social Networks 11, pp. 1 – 37. conversations related to this work, and Ed Curtis at NETL for [27] Wasserman, S. and K. Faust, 1994. Social Network Analysis: Methods his contribution to the ongoing studies of network and Applications, Cambridge University Press, 825 pp. vulnerability in the United States. [28] Fradi, A., S. Brignone, B. Wollenberg, 2001. “Calculation of energy transaction allocation factors,” IEEE Transactions on Power Systems 16:2, pp. 266 – 272. VII. REFERENCES [29] Freeman, L. C., S. P. Borgatti, and D. R. White, 1991. “Centrality in [1] Carreras, B.A., D.E. Newman, I. Dobson, and A.B. Poole, 2004. Valued Graphs: A Measure of Betweenness Based on Network Flow,” “Evidence for Self-Organized Criticality in a Time Series of Electric Social Networks 13, pp. 141 – 154. [30] Brandes, U. and D. Fleischer, 2005. “Centrality Measures Based on Power System Blackouts,” IEEE Transactions on Circuits and Systems nd 51:9, pp. 1733 – 1740. Current Flow,” Proceedings of the 22 Symposium on Theoretical [2] Hines, P., J. Apt, H. Liao, and S. Talukdar, 2006. “The Frequency of Aspects of Computer Science, pp. 533 – 544. Large Blackouts in the United States Electrical Transmission System: [31] Wu, F., Y. Ni, and P. Wei, 2000. “Power Transfer Allocation for Open An Empirical Study,” presented at the 2nd Carnegie Mellon Conference Access Using Graph Theory – Fundamentals and Applications in on Electric Power, January 2006, available from Systems Without Loop Flow,” IEEE Transactions on Power Systems http://www.ece.cmu.edu/~electricityconference/Old06/. 15:3, pp. 923 – 929. [3] Joskow, P., 2003. “The Blackout,” working paper, available at [32] Vajjhalla, S. and P. Fischbeck, 2006. “Quantifying Siting Difficulty: A http://econ-www.mit.edu/faculty/index.htm?prof_id=pjoskow&type=pap Case Study of U.S. Transmission Line Siting,” Energy Policy, er. forthcoming. [4] Casazza, J., F. Delea, and G. Loehr, 2007. “Electrons vs. People: Which [33] Energy Policy Act of 2005, Sec. 1221 is Smarter?” IEEE Power and Energy 5:1. [34] Albert, R., H. Jeong, and A.-L. Barabasi, 2000. “Error and attack [5] Newman, M. E. J., 2003. “The Structure and Function of Complex tolerance of complex networks.” Nature, v. 406, pp. 378-382. Networks,” SIAM Review 45:2, pp. 167 – 256. [35] L. Garber, “Denial-of-service attacks rip the internet,” Computer (IEEE), [6] Albert, R. and A. Barabasi, 2002. “Statistical Mechanics of Complex v. 33, pp.12-17, 2000. Networks,” Reviews of Modern Physics 74, pp. 47 – 97. [7] Erdos, P. and A. Renyi, 1959. “On Random Graphs,” Publ. Math. VIII. BIOGRAPHIES Debrecen 6, pp. 290 – 297. [8] Milgrom, S., 1967. “The Small World Problem,” Psychology Today 2, pp. 60 – 67. Seth Blumsack (M ’06) is an Assistant Professor in the Dept. of Energy and [9] Watts, D. and H. Strogatz, 1998. “Collective Dynamics of ‘Small- Mineral Engineering at Pennsylvania State University, University Park, PA. World’ Networks, Nature 393, pp. 440 – 442. He received the M.S. degree in Economics from Carnegie Mellon in 2003 and [10] Barabasi, A. and R. Albert, 1999. “Emergence of Scaling in Random the Ph.D. in Engineering and Public Policy from Carnegie Mellon in 2006. Networks,” Science 286, pp. 509 – 512. His main interests are regulation and deregulation of network industries, [11] Amaral, L., A. Scala, M. Barthelemy, and H. Stanley, 2000. “Classes of transmission planning and pricing, and complex networks and systems, Small-World Networks,” Proceedings of the National Academy of especially for energy and electric power. Sciences 91:21, pp. 11149 – 11152. [12] 2003 EPRI Technology Roadmap, available at Paul Hines is an Assistant Professor in the School of Engineering at the http://www.epri.com/roadmap/. University of Vermont (UVM). He received the Ph.D. degree in Engineering [13] Blumsack S., L. B. Lave, and M. Ilic, 2006. “A Quantitative Analysis of and Public Policy from Carnegie Mellon University in 2007 and the M.S. the Relationship Between Congestion and Reliability in Electric Power degree in Electrical Engineering from the U. of Washington in 2001. Networks,” Energy Journal, forthcoming. Formerly he worked Research Scientist at the National Energy Technology [14] Blumsack, S., M. Ilic, and L. B. Lave, 2006. “Decomposing Congestion Laboratory, for the Federal Energy Regulatory Commission, where he studied and Reliability,” Carnegie Mellon Electricity Industry Center working interactions between nuclear power plants and transmission networks, Alstom paper CEIC-06-10. ESCA, where he designed a short-term load forecasting tool and for Black and [15] P. Hines, “Model Predictive Control and Artificial Reciprocal Altruism: Veatch, where he worked on substation design projects. His main research A Decentralized Method for Reducing the Social Costs Associated with interests are in the areas of complex systems and networks, the control of Cascading Failures in Electrical Power Networks,” Ph.D. dissertation, cascading failures in power systems and energy system reliability. Carnegie Mellon University, 2007.
[16] Albert, R., I. Albert, and G. Nakarado, 2004. “Structural Vulnerability of the North American Power Grid,” Physical Review E 69, 025103. [17] Chassin, D. and C. Posse, 2005. “Evaluating North American Electric Grid Reliaiblity Using the Barabasi-Albert Network Model,” Physica A 355, pp. 667 – 677. [18] Holmgren, A., 2006. “Using Graph Models to Analyze the Vulnerability of Electric Power Networks,” Risk Analysis 26:4, pp. 955 – 969. [19] Crucitti, P., V. Latora and M. Marchiori, 2004. “Topological Analysis of the Italian Electric Power Grid,” Physica A 338, pp. 92 – 97. [20] [20] Rosato, V., S. Bologna and F. Tiriticco, 2006. “Topological Properties of High-Voltage Electrical Transmission Networks,” Electric Power Systems Research, in press. [21] Lesieutre, B. and I. Hiskens, 2005. “Convexity of the Set of Feasible Injections and Revenue Adequacy in FTR Markets,” IEEE Transactions on Power Systems 20, pp. 1790 – 1798.